Question

evaluate the following integral or state that it diverges.
int_{3}^{infty} \frac{3 dy}{yln y}
select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. the integral converges and (int_{3}^{infty} \frac{3 dy}{yln y}=). (type an exact answer.)
b. the integral diverges.

Explanation:

Step1: Use substitution

Let $u = \ln y$, then $du=\frac{1}{y}dy$. When $y = 3$, $u=\ln 3$; as $y
ightarrow\infty$, $u
ightarrow\infty$. The integral $\int_{3}^{\infty}\frac{3}{y\ln y}dy$ becomes $3\int_{\ln 3}^{\infty}\frac{1}{u}du$.

Step2: Evaluate the integral

The antiderivative of $\frac{1}{u}$ is $\ln|u|$. So, $3\int_{\ln 3}^{\infty}\frac{1}{u}du=3\lim_{b
ightarrow\infty}\int_{\ln 3}^{b}\frac{1}{u}du$.
$=3\lim_{b
ightarrow\infty}(\ln u|_{\ln 3}^{b})$.
$=3\lim_{b
ightarrow\infty}(\ln b-\ln(\ln 3))$.

Step3: Determine convergence or divergence

As $b
ightarrow\infty$, $\lim_{b
ightarrow\infty}(\ln b-\ln(\ln 3))=\infty$. So, $3\lim_{b
ightarrow\infty}(\ln b - \ln(\ln 3))=\infty$.

Answer:

B. The integral diverges.